# total integral closure

A commutative^{} unitary ring $R$ is said to be *totally integrally closed ^{}* if it does not have an overring which is both an integral and an essential extension of $R$.

All totally integrally closed rings are reduced (http://planetmath.org/ReducedRing).

Suppose that $R$ is any commutative ring and that $\overline{R}$ is an integral and essential extension of $R$. If $\overline{R}$ is a totally integrally closed ring, then $\overline{R}$ is called a *total integral closure* of $R$.

For fields the concept totally integrally closed, integrally closed and algebraically closed^{} coincide.

Let $A$ be an integral domain^{}, then its total integral closure is the integral closure^{} of $A$ in the algebraic closure of $\mathrm{Quot}(A)$.

Enochs has first proven that all commutative reduced rings have total integral closure and this is unique up to ring isomorphism.

Title | total integral closure |
---|---|

Canonical name | TotalIntegralClosure |

Date of creation | 2013-03-22 18:51:44 |

Last modified on | 2013-03-22 18:51:44 |

Owner | jocaps (12118) |

Last modified by | jocaps (12118) |

Numerical id | 14 |

Author | jocaps (12118) |

Entry type | Definition |

Classification | msc 13B22 |

Defines | total integral closure |